Abstract

For a wave equation with pure delay, we study an inhomogeneous initial-boundary value problem in a bounded 1D domain.
Under smoothness assumptions, we prove unique existence of classical solutions for any given finite time horizon
and give their explicit representation.
Continuous dependence on the data in a weak extrapolated norm is also shown.

1 Introduction

The wave equation is a typical linear hyperbolic second-order partial differential equation
which naturally arises when modeling phenomena of continuum mechanics such as
sound, light, water or other kind of waves in acoustics, (electro)magnetics, elasticity and fluid dynamics, etc.
(cf. [6, 13]).
Providing a rather adequate description of physical processes,
partial differential equations, or equations with distributed parameters in general,
have found numerous applications in mechanics, medicine, ecology, etc.
Introducing after-effects such as delay into these equations has gained a lot of attention
over several past decades. See e.g., [2, 3, 7, 8].
Mathematical treatment of such systems requires additional carefulness since distributed systems
with delay often turn out to be even ill-posed (cf. [4, 5, 12]).

In the present paper, we consider an initial-boundary value problem for a general linear wave equation
with pure delay and constant coefficients in a bounded interval subject to non-homogeneous Dirichlet boundary conditions.
To solve the equation, we employ Fourier’s separation method as well as the special functions
referred to as delay sine and cosine functions
which were introduced in [9, 10].
We prove the existence of a unique classical solution on any finite time interval,
show its continuous dependence on the data in a very weak extrapolated norm,
give its representation as a Fourier series and prove its absolute and uniform convergence
under certain conditions on the data.

2 Equation with pure delay

For T>0, l>0, we consider the following linear wave equation in a bounded interval (0,l) with a single delay
being a second order partial difference-differential equation for an unknown function η

∂ttη(t,x)=a2∂xxη(t−τ,x)+b∂xη(t−τ,x)+dη(t−τ,x)+g(t,x) for (t,x)∈(0,T)×(0,l)

η(t,0)=θ1(t),η(t,l)=θ2(t) for t>−τ,η(t,x)=ψ(t,x) for (t,x)∈(−τ,0)×(0,l).

(2.2)

Since we are interested in studying classical solutions,
the following compatibility conditions are required to assure for smoothness of the solution on the boundary of time-space cylinder

ψ(t,0)=θ1(t),ψ(t,l)=θ2(t) for t>−τ.

Definition 2.1.

Under a classical solution to the problem (2.1), (2.2) we understand
a function η∈C0([−τ,T]×[0,l]) which satisfies
∂ttη,∂txη,∂xxη∈C0([−τ,0]×[0,l])
as well as
∂ttη,∂txη,∂xxη∈C0([0,T]×[0,l])
and, being plugged into Equations (2.1), (2.2), turns them into identity.

Remark 2.2.

The previous definition does not impose any continuity of time derivatives in t=0.
If the continuity or even smoothness are desired,
additional compatibility conditions on the data, including g, are required.

Let ∥⋅∥k,2:=∥⋅∥Hk,2((0,l)), k∈N0, denote the standard Sobolev norm (cf. [1])
and ∥⋅∥−k,2:=∥⋅∥H−k,2((0,l)) denote the norm of corresponding negative Sobolev space.
We introduce the norm ∥⋅∥X:=√∞∑k=0∥⋅∥2−k,2
and define the Hilbert space X as a completion of L2((0,l)) with respect to ∥⋅∥X.
Obviously, X↪(D((0,l)))′,
i.e., X can be continuously embedded into the space of distributions.

With this notation, we easily see that A:=a2∂2x+b∂x+d
(with ∂x denoting the distributional derivative)
is a bounded linear operator on X since

Corollary 2.4.

Remark 2.5.

It was essential to consider the weak space X.
If the space corresponding to the usual wave equation is used, i.e., (η,ηt)∈H10((0,l))×L2((0,l)),
there follows from [5]
that Equation (2.1), (2.2) is an ill-posed problem due to the lack of continuous dependence
on the data even in the homogeneous case.

Next, we want to establish conditions on the data allowing for the existence of a classical solution.
Performing the substitution

ξ(t,x):=e−b2a2xη(t,x) for (t,x)∈[−τ,T]×[0,l]

(2.8)

with a new unknown function ξ (cp. [11]), the initial boundary value problem (2.1), (2.2)
can be written in the following simplified form with a self-adjoint operator on the right-hand side

∂ttξ(t,x)=a2∂xxξ(t−τ,x)+cξ(t−τ,x)+f(t,x) for (t,x)∈(0,T)×(0,l)

(2.9)

with c:=d−b24a2 complemented by the following boundary and initial conditions

ξ(t,0)

=μ1(t),ξ(t,l)=μ2(t) for t>−τ with μ1(t):=θ1(t),μ2(t):=eb2a2lθ2,

(2.10)

ξ(t,x)

=φ(t,x) for (t,x)∈(−τ,0)×(0,l) with % φ(t,x):=eb2a2xψ(t,x)

(2.11)

and

f(t,x):=eb2a2xg(t,x) for (t,x)∈[0,T]×[0,l].

The solution will be determined in the form

ξ(t,x)=ξ0(t,x)+ξ1(t,x)+G(t,x).

Here, G is an arbitrary function with ∂ttG,∂txG,∂xxG∈C0([−τ,T]×[0,l])
satisfying the boundary conditions

3 Homogeneous equation

In this section, we obtain a formal solution to the initial-boundary value problem (2.13)
with initial and boundary conditions given in Equations (2.10), (2.11).
We exploit Fourier’s separation method to determine ξ0 in the product form ξ0(t,x)=T(t)X(x).
After plugging this ansatz into Equation (2.13), we find

X(x)¨T(t)=a2X′′(x)T(t−τ)+cX(x)T(t−τ).

Hence,

X(x)(¨T(t)−cT(t−τ))=a2X′′(x)T(t−τ).

By formally separating the variables, we deduce

X′′(x)X(x)=¨T(t)−cT(t−τ)a2T(t−τ)=−λ2.

Thus, the equation can be decoupled as follows

¨T(t)+(a2λ2−c)T(t−τ)=0,X′′(x)+λ2X(x)=0.

(3.1)

These are linear second order ordinary (delay) differential equations with constant coefficients.

Due to the zero boundary conditions for ξ0, the boundary conditions for the second equation in (3.1) will also be homogeneous, i.e.,

X(0)=0,X(l)=0.

Therefore, we obtain a Sturm & Liouville problem admitting non-trivial solutions only for the eigennumbers

The initial conditions for each of the equations in (3.2) can be obtained by expanding the initial data
into a Fourier series with respect to the eigenfunction basis of the second equation in (3.1)

(3.3)

for t∈[−τ,T].
Let us further determine the solution of the Cauchy problem associated with each of the equations in (3.2)
subject to the initial conditions from Equation (3.3).

First, we briefly present some useful results from the theory of second order delay differential equations
with pure delay obtained in [9].
The authors considered a linear homogeneous second order delay differential equation

¨x(t)+ω2x(t−τ)=0 for t∈(0,∞),x(t)=β(t) for t∈[−τ,0].

(3.4)

They introduced two special functions referred to as delay cosine and sine functions.
Exploiting these functions, a unique solution to the initial value problem (3.4) was obtained.

Definition 3.1.

Delay cosine is the function given as

(3.5)

with 2k-order polynomials on each of the intervals (k−1)τ≤t<kτ
continuously adjusted at the nodes t=kτ, k∈N0.

Definition 3.2.

with (2k+1)-order polynomials on each of the intervals (k−1)τ≤t<kτ
continuously adjusted at the nodes t=kτ, k∈N0.

Figure 2: Delay sine function

There has further been proved that delay cosine uniquely solves the linear homogeneous second order ordinary delay differential equation
with pure delay subject to the unit initial conditions x≡1 in [−τ,0]
and the delay sine in its turn solves Equation (3.4) subject to the initial conditions
x(t)=ω(t+τ) for t∈[−τ,0].

Using the fact above, the solution of the Cauchy problem was represented in the integral form.
In particular, the solution x to the homogeneous delay differential equation (3.4)
with the initial conditions x≡β in [−τ,0] for an arbitrary β∈C2([−τ,0])
was shown to be given as

x(t)=β(−τ)cosτ(ω,t)+1ω˙β(−τ)sinτ(ω,t)+1ω∫0−τsinτ(ω,t−τ−s)¨β(s)ds.

(3.7)

Turning back to the delay differential equation (3.2) with the initial conditions (2.4),
we obtain their unique solution in the form

Thus, assuming sufficient smoothness of the data to be specified later,
the solution ξ0 to the homogeneous equation (2.13) satisfying homogeneous boundary and
non-homogeneous initial conditions ξ≡Φ in [−τ,0]×[0,l] reads as

ξ0(t,x)=∞∑n=1(Φn(−τ)cosτ(ωn,t)+1ωn˙Φn(−τ)sinτ(ωn,t)+1ωn∫0−τsinτ(ωn,t−τ−s)¨Φn(s)ds)sin(πnlx),Φn(t)=2l∫l0(φ(t,s)−G(t,s))sin(πnls)ds for n∈N.

(3.9)

4 Non-homogeneous equation

Next, we consider the non-homogeneous equation (2.16) with
the right-hand side from Equation (2.18) subject to homogeneous initial and boundary conditions

Then the classical solution to problem (2.9)–(2.11) can be represented as an absolutely and uniformly convergent
Fourier series given in Equation (5.1).
The latter series is a twice continuously differentiable function with respect to both variables.
Its derivatives of order less or equal two with respect to t and x can be obtained by a term-wise differentiation
of the series and the resulting series are also absolutely and uniformly convergent in [0,T]×[0,l].

holds true, the series S1 as well as its derivatives of order less or equal 2 converge absolutely and uniformly.
Note that a single differentiation with respect to x corresponds, roughly speaking, to a multiplication with n.

Next, we consider the coefficients Bn.
For an arbitrary t∈[0,T] with
(k−1)τ≤t<kτ, we perform the substitution t−τ−s=ξ and exploit the mean value theorem to estimate

Applying the theorem on differentiation under the integral sign to Bn and taking into account that
sinτ(πnla,⋅) is twice weakly differentiable in [0,∞), namely: sinτ(πnla,⋅)∈W2,∞loc((0,∞)),
its derivatives are polynomials of order lower than those of sinτ(πnla,⋅)
and their convolution with ¨Φn is continuous,
analogous estimates can be obtained for ˙Φn and ¨Φn which, in their turn,
also follow to be continuous functions.

Now, if the condition

limn→∞maxs∈[−τ,0]|¨Φn|n2m+3+α=0

is satisfied, the series S2 as well as its derivatives of order less or equal 2 converge absolutely and uniformly.

Finally, we look at the Fourier coefficients Cn.
Again, for an arbitrary period of time t∈[0,T] with (k−1)t≤t<kτ, 0≤k≤m,
we substitute t−τ−ξ=s.
Once again, using the mean value theorem, we estimate

As before, Cn can be shown to be twice continuously differentiable.
If now

limn→∞maxk=1,…,mmaxt∈[(k−1)τ,max{kτ,T}]|Fn(t)|n2k+3+α=0

(5.3)

is satisfied, then both S3 and its derivatives of order less or equal 2 converge absolutely and uniformly.

Since all three conditions are guaranteed by the assumptions of the Theorem due to the fact k≤m,
the proof is finished.
∎

Remark 5.2.

From the practical point of view, the rapid decay condition on the Fourier coefficients of the data given in Equation (5.2)
mean a sufficiently high Sobolev regularity of the data and corresponding higher order
compatibility conditions at the boundary of (0,l) (cf. [11]).